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Lang."],"published-print":{"date-parts":[[2026,8,17]]},"abstract":"<jats:p>\n                    We introduce a new programming language and its categorical semantics in order to design and implement neural networks within the framework of algebraic effects and handlers for arrows. Our language enables us to construct neural networks symbolically, in the same manner as algebraic effects, and to assign implementations\u2014such as backpropagation computations\u2014to them via handlers. The advantage of this language design is that network descriptions become abstract and high-level, while implementations can be flexibly assigned to networks. We establish a rigorous foundation for our language by developing a type system, an operational semantics, a categorical semantics, and soundness and adequacy theorems. The technical core is the introduction of\n                    <jats:italic toggle=\"yes\">reverse handlers<\/jats:italic>\n                    , a novel handler mechanism for arrows for implementing backpropagation, together with new algebras of strong promonads on reverse differential restriction categories (RDRCs), whose string diagrams provide a formal graphical syntax and semantics for neural networks.\n                  <\/jats:p>","DOI":"10.1145\/3828704","type":"journal-article","created":{"date-parts":[[2026,8,17]],"date-time":"2026-08-17T20:23:46Z","timestamp":1786998226000},"page":"921-949","update-policy":"https:\/\/doi.org\/10.1145\/crossmark-policy","source":"Crossref","is-referenced-by-count":0,"title":["Programming Backpropagation with Reverse Handlers for Arrows"],"prefix":"10.1145","volume":"10","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3409-6963","authenticated-orcid":false,"given":"Takahiro","family":"Sanada","sequence":"first","affiliation":[{"name":"Fukui Prefectural University, Eiheiji, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0007-7287-7572","authenticated-orcid":false,"given":"Keisuke","family":"Hoshino","sequence":"additional","affiliation":[{"name":"Kyoto University, Kyoto, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-1272-7684","authenticated-orcid":false,"given":"Kenshin","family":"Hirai","sequence":"additional","affiliation":[{"name":"Kyoto University, Kyoto, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-7529-5489","authenticated-orcid":false,"given":"Shin-ya","family":"Katsumata","sequence":"additional","affiliation":[{"name":"Kyoto Sangyo University, Kyoto, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"320","published-online":{"date-parts":[[2026,8,17]]},"reference":[{"key":"e_1_3_1_2_2","first-page":"265","volume-title":"Proceedings of the 12th USENIX Conference on Operating Systems Design and Implementation (Savannah, GA, USA) (OSDI\u201916)","author":"Abadi Mart\u00edn","year":"2016","unstructured":"Mart\u00edn Abadi, Paul Barham, Jianmin Chen, Zhifeng Chen, Andy Davis, Jeffrey Dean, Matthieu Devin, Sanjay Ghemawat, Geoffrey Irving, Michael Isard, Manjunath Kudlur, Josh Levenberg, Rajat Monga, Sherry Moore, Derek G. 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